Szabolcs Zakany

The skin color patterning of lizards with monochromatic skin scales can be modeled computationally with at least two different mathematical frameworks: Turing's reaction-diffusion model (relying on interaction data among colored cells), and the Lenz-Ising model (based on effective mesoscopic interactions among individual skin scales). However, a mechanistic connection between these very different descriptions remains elusive. Capitalizing both on the biological interpretability of the reaction-diffusion model and on the geometric interpretability of the Lenz-Ising model, we develop a computational approach to bridge this gap in a series of evolutionary divergent lizards. First, by combining mathematical and computational methods to perform a mapping between the two models, we show that a four-parameter extension of the classic Lenz-Ising model efficiently captures statistical aspects of color patterning in all species. Second, we show that the Lenz-Ising model parameters acquire an effective interpretation, in terms of cell interactions, by observing how these parameters change when tuning the reaction-diffusion parameters optimized for each species studied here. Taken together, our findings establish quantitative links between the strength and length scales of microscopic interactions among colored cells, and the scale-by-scale lizard skin color patterns visible to the naked eye. Published by the American Physical Society 2025

Turing proposed a reaction-diffusion (RD) process as the chemical basis of morphogenesis. Despite the elegance of this model, its relevance for the precise description of morphogenesis in real organisms is largely disputed. Here, we show that a simple RD system, predicting the cellular-automaton-like patterning of ocellated lizards’ skin into green and black labyrinthine chains of scales, additionally predicts unsuspected subtle color subclustering that correlates with the colors of the scales’ neighbors. Hyperspectral imaging indicates that color subclustering is present in real lizards, confirming the numerical model nontrivial prediction. In addition, extensive histological analyses show that melanophores’ spatial distribution correlates with scale neighborhood, confirming that color subclustering is associated to the underlying microscopic system of chromatophore interactions. We then show that the observed subclustering is efficiently captured by RD models, irrespective of their form, discretization, and spatial dimensionality. We also show that sets of values can be identified in the 12-dimensional RD parameter space to yield the correct direction of correlation (i.e., observed in real lizards) between green-scale blackness and their neighborhood configuration, hence instructing the mathematical model. More generally, our results show that subtle mesoscopic properties of biological dynamical systems, as well as some of the underlying microscopic features, are quantitatively captured by simple RD models without integrating the unmanageable profusion of variables at lower scales.

Two influential concepts in tissue patterning are Wolpert’s positional information and Turing’s self-organized reaction–diffusion (RD). The latter establishes the patterning of hair and feathers. Here, our morphological, genetic, and functional—by CRISPR-Cas9–mediated gene disruption—characterization of wild-type versus “scaleless” snakes reveals that the near-perfect hexagonal pattern of snake scales is established through interactions between RD in the skin and somitic positional information. First, we show that ventral scale development is guided by hypaxial somites and, second, that ventral scales and epaxial somites guide the sequential RD patterning of the dorsolateral scales. The RD intrinsic length scale evolved to match somite periodicity, ensuring the alignment of ribs and scales, both of which play a critical role in snake locomotion.

The ocellated lizard (Timon lepidus) exhibits an intricate skin color pattern made of monochromatic black and green skin scales, whose dynamics of color flipping are known to be well modeled by a stochastic cellular automaton. We show that the late-time probability distribution of the pattern corresponds to the canonical probability distribution of the antiferromagnetic Ising model and can be generated by dynamics different from the commonly-used Glauber. We comment on skin scale patterns generated by the Ising model on the triangular lattice in the low-temperature limit.

It has been recently conjectured that the exact eigenfunctions of quantum mirror curves can be obtained by combining their WKB expansion with the open topological string wavefunction. In this paper we give further evidence for this conjecture. We present closed expressions for the wavefunctions in the so-called maximally supersymmetric case, in various geometries. In the higher genus case, our conjecture provides a solution to the quantum Baxter equation of the corresponding cluster integrable system, and we argue that the quantization conditions of the integrable system follow from imposing appropriate asymptotic conditions on the wavefunction. We also present checks of the conjecture for general values of the Planck constant.

Topological strings on toric Calabi-Yau threefolds can be defined non-perturbatively in terms of a non-interacting Fermi gas of N particles. Using this approach, we propose a definition of quantum mirror curves as quantum distributions on phase space. The quantum distribution is obtained as the Wigner transform of the reduced density matrix of the Fermi gas. We show that the classical mirror geometry emerges in the strongly coupled, large N limit in which ℏ ∼ N. In this limit, the Fermi gas has effectively zero temperature, and the Wigner distribution becomes sharply supported on the interior of the classical mirror curve. The quantum fluctuations around the classical limit turn out to be captured by an improved version of the universal scaling form of Balazs and Zipfel.

Topological strings on toric Calabi--Yau threefolds can be defined non-perturbatively in terms of a free Fermi gas of N particles. Using this approach, we propose a definition of quantum mirror curves as quantum distributions on phase space. The quantum distribution is obtained as the Wigner transform of the reduced density matrix of the Fermi gas. We show that the classical mirror geometry emerges in the strongly coupled, large N limit in which hbar ~ N. In this limit, the Fermi gas has effectively zero temperature, and the Wigner distribution becomes sharply supported on the interior of the classical mirror curve. The quantum fluctuations around the classical limit turn out to be captured by an improved version of the universal scaling form of Balazs and Zipfel.